Optimal. Leaf size=87 \[ \frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {d x^2 (b c-a d)^2}{2 b^3}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b} \]
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Rubi [A] time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \[ \frac {d x^2 (b c-a d)^2}{2 b^3}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {\left (c+d x^2\right )^3}{6 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^3}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {d (b c-a d)^2 x^2}{2 b^3}+\frac {(b c-a d) \left (c+d x^2\right )^2}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.94 \[ \frac {b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 120, normalized size = 1.38 \[ \frac {2 \, b^{3} d^{3} x^{6} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 124, normalized size = 1.43 \[ \frac {2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 149, normalized size = 1.71 \[ \frac {d^{3} x^{6}}{6 b}-\frac {a \,d^{3} x^{4}}{4 b^{2}}+\frac {3 c \,d^{2} x^{4}}{4 b}+\frac {a^{2} d^{3} x^{2}}{2 b^{3}}-\frac {3 a c \,d^{2} x^{2}}{2 b^{2}}+\frac {3 c^{2} d \,x^{2}}{2 b}-\frac {a^{3} d^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}-\frac {3 a \,c^{2} d \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {c^{3} \ln \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 119, normalized size = 1.37 \[ \frac {2 \, b^{2} d^{3} x^{6} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 123, normalized size = 1.41 \[ x^2\,\left (\frac {3\,c^2\,d}{2\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b^4}+\frac {d^3\,x^6}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 94, normalized size = 1.08 \[ x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{2} \left (\frac {a^{2} d^{3}}{2 b^{3}} - \frac {3 a c d^{2}}{2 b^{2}} + \frac {3 c^{2} d}{2 b}\right ) + \frac {d^{3} x^{6}}{6 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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